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A generalized Bernstein approximation theorem

Published online by Cambridge University Press:  24 October 2008

João B. Prolla
João B. Prolla
Affiliation:
Departamento de Matemática – IMECC, Universidade Estadual de Campinas, Caixa Postal 6065, 13081 – Campinas, S.P., Brazil
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Extract

A celebrated theorem of Weierstrass states that any continuous real-valued function f defined on the closed interval [0, 1] ⊂ ℝ is the limit of a uniformly convergent sequence of polynomials. One of the most elegant and elementary proofs of this classic result is that which uses the Bernstein polynomials of f

one for each integer n ≥ 1. Bernstein's Theorem states that Bn(f) → f uniformly on [0, 1] and, since each Bn(f) is a polynomial of degree at most n, we have as a consequence Weierstrass' theorem. See for example Lorentz [9]. The operator Bn, defined on the space C([0, 1]; ℝ) with values in the vector subspace of all polynomials of degree at most n has the property that Bn(f) ≥ 0 whenever f ≥ 0. Thus Bernstein's Theorem also establishes the fact that each positive continuous real-valued function on [0, 1] is the limit of a uniformly convergent sequence of positive polynomials. This raises the following natural question: consider a compact Hausdorff space X and the convex cone C+(X):= {f ∈ C(X; ℝ); f ≥ 0}. Now the analogue of Bernstein's Theorem would be a theorem stating when a convex cone contained in C+(X) is dense in it. More generally, one raises the question of describing the closure of a convex cone contained in C(X; ℝ), and, in particular, the closure of A+:= {f ∈ A; f ≥ 0}, where A is a subalgebra of C(X; ℝ).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 104 , Issue 2 , September 1988 , pp. 317 - 330
Copyright
Copyright © Cambridge Philosophical Society 1988

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References

REFERENCES

[1]Bishop, E.. A generalization of the Stone–Weierstrass theorem. Pacific J. Math. 11 (1961), 777–783.CrossRefGoogle Scholar
[2]Bonsall, F. F.. Semi-algebras of continuous functions. Proc. London Math. Soc. (3) 10 (1960), 122–140.CrossRefGoogle Scholar
[3]Brosowski, B. and Deutsch, F.. An elementary proof of the Stone–Weierstrass theorem. Proc. Amer. Math. Soc. 81 (1981), 89–92.CrossRefGoogle Scholar
[4]Burckel, R. B.. Bishop's Stone–Weierstrass theorem. Amer. Math. Monthly 91 (1984), 22–32.Google Scholar
[5]Ellis, A. J.. Some approximation results for function spaces. Indag. Math. 42 (1980), 125–130.CrossRefGoogle Scholar
[6]Feyel, D. and de La Pradelle, . Sur certaines extensions du Théorème d'approximation de Bernstein. Pacific J. Math. 115 (1984), 81–89.CrossRefGoogle Scholar
[7]Glicksberg, I.. Measures orthogonal to algebras and sets of antisymmetry. Trans. Amer. Math. Soc. 104 (1962), 415–435.CrossRefGoogle Scholar
[8]Jewett, R. I.. A variation on the Stone–Weierstrass theorem. Proc. Amer. Math. Soc. 14 (1963), 690–693.CrossRefGoogle Scholar
[9]Lorentz, G. G.. Approximation of Functions (Holt, Rinehart and Winston, 1966).Google Scholar
[10]Machado, S.. On Bishop's generalization of the Weierstrass–Stone theorem. Indag. Math. 39 (1977), 218–224.CrossRefGoogle Scholar
[11]Nachbin, L.. Weighted approximation for algebras and modules of continuous functions: real and self-adjoint complex cases. Ann. of Math. 81 (1965), 289–302.CrossRefGoogle Scholar
[12]Nachbin, L.. Elements of Approximation Theory (van Nostrand, 1967; reprinted, Krieger, 1976).Google Scholar
[13]von Neumann, J.. Probabilistic logics and the synthesis of reliable organisms from unreliable components. In Automata Studies (Princeton University Press, 1956), pp. 93–94.Google Scholar
[14]Prolla, J. B.. Approximation of Vector Valued Functions (North-Holland, 1977).Google Scholar
[15]Ransford, T. J.. A short elementary proof of the Bishop–Stone–Weierstrass theorem. Math. Proc. Cambridge Philos. Soc. 96 (1984), 309–311.CrossRefGoogle Scholar
[16]Shilov, G. E.. On rings of functions with uniform convergence. Ukrain. Mat. Zh. 3 (1951), 404–411.Google Scholar